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Random Utility Model Individual is utility from a choice j can be decomposed into two components: V ij is deterministic – common to everyone, given the same characteristics and constraints – representative tastes of the population e.g. effects of time and cost on travel mode choice ij is random –reflects idiosyncratic tastes of i and unobserved attributes of choice j

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Random Utility Model V ij is a function of attributes of alternative j (e.g. price and time) and observed consumer and choice characteristics. We are interested in finding,, Lets forget about z now for simplicity

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RUM and binary choices Consider two choices e.g. bus or car We observe whether an individual uses one or the other Define What is the probability that we observe an individual choosing to travel by bus? Assume utility maximisation Individual chooses bus (y=1) rather than car (y=0) if utility of commuting by bus exceeds utility of commuting by car

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RUM and binary choices So choose bus if So the probability that we observe an individual choosing bus travel is

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The linear probability model Assume probability depends linearly on observed characteristics (price and time) Then you can estimate by linear regression Where is the dummy variable for mode choice (1 if bus, 0 if car) Other consumer and choice characteristics can be included (the zs in the first slide in this section)

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The linear probability model Unfortunately his has some undesirable properties 1 0 Linear regression line

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Multiple choices We often want to think about many more than two choices –Choice of regional location –Choice of transport mode with many alternatives –Choice amongst a sample of schools How can we extend the binary choice logit model? Random Utility model extends to many choices Choose choice k if utility higher than for all other choices

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Multinomial logit (1) Again we need to assume some distribution for the unobserved factor One type of distribution (extreme value) gives a simple solution for the probability that choice k is made: This is a generalisation of the logit model with many alternatives = multinomial logit or conditional logit

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Multinomial logit (2) Recall: V ij is a linear function of observed characteristics of the individuals and their choices. e.g. for travel mode choice Parameters estimated: For an individual characteristic that is common across choices (e.g. income, gender): one parameter per choice –For at least one choice this is zero (base case). For a characteristic which varies only across choices e.g. price of transport: one parameter common across choices

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Example: Value of time MNL models used to estimate value of travel time with from observed commuter behaviour Three transport choices: bus (0), train (1), car (2) Choosing bus as the base case:

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The independence of irrelevant alternatives problem (IIA) and the nested logit model

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Multinomial logit and IIA Many applications in economic and geographical journals (and other research areas) The multinomial logit model is the workhorse of multiple choice modelling in all disciplines. Easy to compute But it has a drawback

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Independence of Irrelevant Alternatives Model assumes that unobserved attributes of all alternatives are perceived as equally similar But will people unable to travel by red bus really switch to travelling by train? Most likely outcome is (assuming supply of bus seats is elastic) –Blue bus: 40% –Train: 60% This failure of multinomial/conditional logit models is called the Independence of Irrelevant Alternatives assumption (IIA)

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Independence of Irrelevant Alternatives It is easy to see why this is: Ratio of probabilities of choosing k (e.g. red bus) and another choice l (e.g. train) is just All other choices drop out of this odds ratio There are models that overcome this, e.g…

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Nested Logit Model Value placed on choices available in second stage (3,4) enter into calculation of choice probabilities in first stage (2)… Logit for bus versus train to estimate V 3 and V 4 Define the Inclusive Value of public transport as Estimate logit model for Car (1) versus Public (2) using:

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Example: Transport mode choice Asensio, J., Transport Mode Choice by Commuters to Barcelonas CBD, Urban Studies, 39(10), 2002 –Some selected coefficients VariableParameter Cost-0.002 Travel time by car-0.054 Travel time by public transport-0.018 Sex (car)0.889 Sex (bus)-1.001 We dont know the units of measurement, but how much more valuable is time saved car than time saved by public transport?

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Micro and aggregated choice models Micro level logit choice models often have aggregated equivalents i.e. if you only have choice characteristics, you could use a choice-level regression of the proportion of individuals making each choice on the choice characteristics Obviously log(n_k) would work too (why?)

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Micro and aggregated choice models In fact, a Poisson model on aggregated data gives exactly the same coefficient estimates as the conditional logit model Which is based on ML estimation of See Guimaraes et al Restats (2003) –though this equivalence was known before this discovery Heres an example…

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Aggregate v micro choice models Hence, theres little point in using conditional logit if you only have choice-characteristics Conditional/multinomial logit is good if you have individual and group-level characteristics The aggregated OLS version gives rise to Spatial interaction models of flows between origins and destinations = Gravity models Widely applied (generally a-theoretically) in migration, trade and commuting applications –e.g. See Head (2003) Gravity for beginners

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Gravity/spatial interaction/migration/trade models Flow from place j to place k modelled as Typically characteristics of destination and source include some measure of attraction e.g. population mass (or market potential in trade models) wages (endogenous) And measure of the cost in moving between place j and d (e.g. log distance) Hence gravity – after Newton

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Strong distance decay effects –Typical elasticities -0.5 to -2.0 Even for internet site visits!: see Blum and Goldfarb (2006) Journal of International Economics Trade literature has many examples Disdier and Head (2003) The Puzzling Persistence Of The Distance Effect On Bilateral Trade, Review of Economics and Statistics –Finds mean distance elasticity of -0.9 from about 1500 studies Gravity/spatial interaction/migration/trade models